A Law of the Iterated Logarithm for Directed Last Passage Percolation

Let ({widetilde{H}}_N), (N ge 1), be the point-to-point last passage times of directed percolation on rectangles ([(1,1), ([gamma N], N)]) in ({mathbb {N}}times {mathbb {N}}) over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some (alpha _{sup } >0),

$$begin{aligned} alpha _{sup } , le , limsup _{N rightarrow infty } frac{{widetilde{H}}_N}{(log log N)^{2/3}} , le , Big ( frac{3}{4} Big )^{2/3} end{aligned}$$

with probability one, and that (alpha _{sup } = big ( frac{3}{4} big )^{2/3}) provided a commonly believed tail bound holds. The result is in contrast with the normalization ((log N)^{2/3}) for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed ((log log N)^{1/3}) is also discussed.